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I've decided to give CK another try.
Once more when my liege asks me to mobilize troops, they just stand there beyond control. Neither can I demobilize them. How is this possible? Is this normal?

When you agree to his mobilization offer they are indeed beyond your control, that they don't move at all can sometimes happen. Save- and reload sometimes fixes this.

When you give your liege control over your troops, you can set army-maintenance to zero, to prevent you from going into debt.

If you don't want your troops to be under the control of your liege and yet you don't want to deny his request, manually mobilize your regiments yourself and then say yes to his request, then you keep control over your own troops.